Investments and …nancial structure with
imperfect …nancial markets: an
intertemporal discrete-time framework
Marco Mazzoli
1Università di Modena e Reggio Emilia
Dipartimento di Economia Politica
Via Berengario 51
41100 Modena
e-mail (priv.): [email protected]
March 28, 2000
Abstract
This paper deals with the problem of simultaneity be-tween the …rm’s investments and …nancial structure, in a context of dynamic optimization, characterised by two main assumptions: …rst of all, diverging incentives for managers and shareholders, secondly, …nancial markets imperfec-tions generating a risk premium on the borrowed …nance. A ”discrete-time” framework has been introduced in order to better model the relevance of tim-ing in the co-ordination process between …nancial and investment decisions, assumed to take place simultaneously. The simple model proposed here may provide some intuitive interpretation for a number of phenomena such as the propagation of …nancial shocks into the real economy and the countercyclical mark-ups.1 Introduction
The relevance of the …rm’s …nancial structure for investments has been the object of a great deal of contributions both in …nance and industrial economics, although the problem of simultaneity between investment and …nancial structure decisions is raised very rarely. In industrial economics, for instance, this issue has been analysed - within the context of predatory pricing models - by the literature on the ”deep pocket argument” (Telser, 1966, Benoît, 1984, Poitervin, 1989a) and by the literature on the assumption of ”limited liability e¤ect” (Brander
1
I am very grateful to Giuseppe Marotta, Sergio Pastorello,
Daniele Ritelli and Kumareswamy Velupillai for their very helpful
comments to a previous draft of this paper. All mistakes are mine.
and Lewis, 1985, Poitervin 1989b). In the former, dominant …rms can a¤ord a long-lasting price war because they can rely on large …nancial resources. In the latter, a high level of debt is regarded as a pre-commitment for an aggressive policy: a signalling game of entry deterrence yields, as a result, the optimal …nancial structure for both the incumbent and the entrant.
In many New-Keynesian literature focusing on the macroeconomic implica-tions of information asymmetries in …nancial markets and agency problems, the …rm’s …nancial structure is often seen as a potential vehicle of di¤usion of in-stability and business cycle, but the attempts to model simultaneous decisions of investment and …nancial structure are extremely rare in an intertemporal context, if one excludes very simpli…ed two-period models.
In intertemporal investment models the use of a constant and exogenous discount rate in investment models with dynamic optimization is, of course, only a convenient algebraic simpli…cation of reality. In fact, recent contributions have introduced assumptions of stochastic behaviour in the interest (see for instance Saltari and Calcagnini, 1995 for a formalization where the discount rate is assumed to follow a Brownian stochastic process) and in …nancial economics models (due to uncertainty and information asymmetries) it is very common to ”correct” the …rm’s cost of …nancial capital for a risk parameter, usually assumed to be a monotonic function of the gearing ratio or the leverage ratio.
However the assumption of simultaneity of investment and choice of …nancial structure generates a number of problems.
First of all, as pointed out by Aoki and Lejonhufvud (1988), the value of the endowment of the physical capital of a …rm is not independent of the en-dowment of physical capital of the competitor …rms. This also means that the concept of market value of physical capital may become somehow ambiguous, and even more so if we think of oligopolistic sectors where the behaviour of the competitors may be modelled by bayesian games. A similar problem could arise even in a perfectly competitive environment if we admit that perfectly rational individuals optimizing the use of information, may have di¤erent expectations of future pro…ts and asset prices, due to their di¤erent theoretical priors, as postulated, for instance by Kurz rational beliefs theory (Kurz, 1994a, 1994b).
Secondly, a great deal of complexity is associated with the presence of agency problems between managers and shareholders.
If we assume that the management has some discretional power in allocating the internally generated cash ‡ow and we further admit that in an imperfectly competitive framework a …rm might have incentive not to reveal the amount of output and pro…ts associated to a certain level of physical capital or, more generally, we admit that the value of its physical capital might be perceived di¤erently by the …rm itself and by its competitors, we might turn out to have some di¢culty in de…ning the concept of expected pro…ts generated by the newly installed capital. In other words, even if the expectations concerning the pro…ts generated by the newly installed capital diverge only in the short run for the …rm and its competitors, exogenous causes a¤ecting the capital gain on the shares (and therefore the cost of internally generated …nance with respect to the borrowed …nance) may a¤ect the …rm’s …nancial structure, its
risk (such as perceived by the external …nancial investors) for long enough to generate an impact on the (adjusted-for-risk) …rm’s rate of discount and level of investments. Furthermore, the length this impact might be associated with rate of capital depreciation, due to .the balance-sheet constraint connecting the existing stock of capital and the various sources of …nance.
In a world of …nancial market imperfections, where the internally generated cash ‡ow constitutes a cheaper source of …nance than borrowing and issuing new shares, the behaviour of the share price and capital gains certainly a¤ects the dividend policy of the management which again a¤ects the …rms …nancial structure by determining the rate of pro…ts retention, which, in its turn , deter-mines the volume of investments …nanced by internal …nance. All that would not have any relevant or persistent e¤ect on the real investments if the share price rapidly adjusted to the net present value of the pro…ts (possibly adjusted to account for the presence of uncertainty and sunk costs) generated by the …rm Our conjecture is, on the other hand, that this could have relevant and persistent e¤ects if we accept that stock prices may diverge for long enough (possibly because of a persistent bubble) from the net present value of future pro…ts. Even more relevant and persistent would be the e¤ects if we accepted that di¤erent groups of rational agents may have divergent expectations, due to their beliefs (as suggested, as we said, by Kurz, 1994a, 1994b).
In Mazzoli (1998, ch. 7) the issue of simultaneity between …nancial structure and investments decision has been informally analysed within a continuous time optimal control framework where the simultaneity of the investments and …nan-cial structure decision generates a functional link among the state and control variables and the rate of discount of future pro…ts. This functional link boils down into a time-dependent Hamiltonian, but the time dependency is removed (although at the cost of some loss of generality) by postulating that the relation-ship between the …rm pro…ts and the share price was a¤ected by another time dependent mechanism of information spreading consistent to the ones informally discussed in Shiller (1989).
Timing in the process of coordination between …nancial and investment de-cisions is essential for the de…nition of ‡ow variables and might not be properly captured by the conventional continuous time models. In order to take into account the relevance of timing we introduce here a recursive structure in the intertemporal problem of the …rm’s investments characterized by the presence of simultaneity between …rms investments and …nancial structure. This models constitute an attempt to formalise a causal link between the ‡ow of pro…ts, the …rm’s optimal choice between own capital (de…ned as non-distributed pro…ts) and borrowing and the rate of discount of the future ‡ows of pro…t. An attempt is also made to remove the possible problems of time dependency by using the implications of the rate of capital depreciation for the duration of each newly installed unit of capital.
The next section contains the description of the model, section 3 contains a proposed approach to solve the problems arising from the time structure of the model, and the last section contains a few concluding remarks.
2 The model
The main assumptions of the present analysis may be summarized as follows: i) The market for goods is assumed to be imperfectly competitive, although perfect competition can be a particular case;
ii) The management is composed of members of the controlling group of the …rm and acts in the interest of the controlling group of the …rm.
The capital is installed at time t ¡ 1, and is …nanced with the …nancial sources raised by the …rm at time t-1 with a contract establishing also their remuneration which will be paid out at time t
At time t the production process takes place, generating the pro…ts ¼t, and
the investment decisions, as well as the payment of the interest on borrowed capital and the dividends on the own capital.are taken. ©¤
t¡1 is the weighted
average of the cost of own capital and borrower capital, established at time t¡ 1 and paid at time t.
Before formalizing the maximand of the intertemporal problem we have to make here a few points. De…ning the the …rm’s intertemporal investment decision over an in…nite horizon would create some di¢culties, since one could not apply the conventional solution approach for discrete-time models, based on backward induction. For this reason we de…ne the …rm’s investment decision problem over a …nite horizon, and, in order to avoid an arbitrary choice of the ending time T , we use the fact that the rate of capital depreciation ± (if de…ned as 1=m, i.e. the reciprocal of an integer m) also implicitly de…nes a time T = 1=± where all the persistent modi…cations on the equilibrium determined by the …rm’s investment and …nancial decisions exhaust their e¤ects. In other words, each and every unit of newly installed capital is also characterized by a physical life implicitly de…ned by its rate of depreciation2.
On the basis of the above assumptions, the problem of the …rm may be represented in the following way.
Vt= 1=± X t=1 ( £ ¼t ¡ kt¡1jº; !l¢¡ It ¤ ¢¡ 1 1 + ©¤ t¡1 ¢t¡1 ) (1)
where ¼tare the pro…ts at time t, kt¡1is the capital installed at time t¡1, It
is the amount of investments (decided at time t that will contribute to determine the stock of capital at time t + 1).
¼t
¡
kt¡1jº ; !l
¢
is assumed to be the pro…ts maximum value function: in other words, the …rm is at any given moment assumed to maximize pro…ts conditional
2Another possibility could have been de…ning the time horizon of the investment decision
according to the decision horizon of the …rm’s management. However, thea approach we have choosen here, by allowing to ”save one assumption”, reduces the degree of the subjectiveness of the model foundations
on the parameter º (describing the market structure and the competitive envi-ronment), and on the labour costs !l, assumed to be given3. In what follows
we will omit both º ; and !l
The maximand 1 is subject to the following constraints 2, 3, 5:
It= kt¡ (1 ¡ ±)kt¡1 (2)
¼t(kt¡1) ¡ ©¤t¡1kt¡1+ ¢Bt+ ¢Et= It (3)
where Btrepresents the borrowed …nance (and, of course, ¢Bt = Bt¡Bt¡1),
Et represents the existing stock of the …rm’s shares, valued at their issue price;
as we said, ± , the rate of capital depreciation, is expressed in the form of 1=m , where m is an integer representing expected life of the newly acquired capital; the non-distributed pro…ts can be de…ned as
¼t(kt¡1) ¡ ©¤t¡1kt¡1= ¢R:
3 is a ‡ow-of-funds condition saying that the new investments Itcan be
…-nanced either by issuing new shares, or by borrowing money or with the residual cash ‡ow which is left after remunerating the …nancial capital (employed to …-nance the physical capital kt). The latter is de…ned as the weighted average
of borrowing and non-distributed pro…ts (i.e. pro…ts in excess to the dividends that the …rms pays out to remunerate the shareholders at the exogenously de-termined yield rs
t). For simplicity, and since the purpose of this paper is to
emphasize the choice between debt and internally generated cash ‡ow, we will assume in what follows that the (exogenously given) time path and variation of the share price and dividends is such as ¢E0 = 0, i.e. the …rm is not issuing
new shares in the periods under consideration.4. Therefore, we will rewrite 3
as follows:
¼t(kt¡1) ¡ ©¤t¡1kt¡1+ ¢Bt= It (4) 3To justify this sort of ”ceteris paribus” assumption we may think of a labour market
characterized by a simpli…ed ”e¢ciency wages” mechanism, where wages and employment are …xed in the short run and are manily a¤ected by macroeconomic factors.
4Introducing the possibility for the …rm to issue new shares would not have modi…ed the
qualitative meaning of this simple discussion. For instance, if we de…ne r¤
s;tas the yield on
the …rm’s shares at time t, ps;tthe share price, ¢ps;tits variation with respect to time t ¡ 1,
Nt the number of existing shares, we could interpret the situation where the …rm issues new
shares as the case where, given D = r¤
s;t¢ ps;t¢ Nt¡ ¢ps;t¢ Nt
we have ¢ps;t¢ Nt> rs;t¤ ¢ ps;t¢ Nt
i.e. a case of ”negative dividends” generated by an extremely favorable valuation of the …rm by the market which makes extremely convenient for the management to raise risk capital.
Furthermore, ©¤ t is de…ned as follows: ©¤t = min ¹t h (1 ¡ ¹t) it+ ¹t ³ rtf+ Á (¹t) ´i (5) where Á (¹t)is the risk premium on the interest rate on the …rm’s borrowing
and it is assumed to be a monotonically increasing function of the gearing ratio ¹t = Bt=kt. ©¤t represents here the minimum value function of the …rm’s
…nancial costs minimization problem: at every time t the …rm is optimizing its …nancial structure by choosing the optimal gearing ratio ¹t = Bt=kt that
minimizes the cost of …nancial capital, de…ned as the weighted average between the borrowed and the internally generated …nance.
The optimized …nancial structure determines the rate of discount appearing in the intertemporal problem. However, the optimal rate of discount is condi-tional on the ‡ow of non-distributed pro…ts of the previous period. In this way the ”…rm-speci…c” rate of discount is recursively determined as a function of the lagged stock of physical capital and lagged cost of …nancial capital.
in 5, itrepresents the cost of the internally generated own capital, de…ned as
follows: it= D E0+ Rt (6) where E0= ps;0¢ Nt and D = r¤s;t¢ ps;t¢ Nt¡ ¢ps;t¢ Nt (7) where r¤
s;t is the yield on the …rm’s shares at time t, ps;t is the share price,
¢ps;tits variation with respect to time t¡1, Ntis the number of existing shares.
Given the (exogenously determined) share price and its short run capital gain, the management of the …rm choose a yield r¤
s;twhich determine the amount of
dividends they want to pay to the shareholders. r¤
s;t may be interpreted as a
…nancial market constraint on the behaviour of the management: in other words it represents the remuneration that they have to grant to the shareholders in order to keep them happy. It could be interpreted as the result of an implicit negotiation between the shareholders and the management and re‡ects the typ-ical principal-agent problem of a …rm where there is asymmetric information between the shareholders and the management. In this regard, we could have two possible benchmarks:
1. the management only pays out to the shareholders the amount of dividends consistent with the market remuneration of the shares;
2. exhaustion of pro…ts into dividends.
The …rst case was considered in Mazzoli (1998, ch. 7) and was meant to describe those kinds of ”non-easily-removable” agency problems between share-holders and managers that may characterise some of the ”bank oriented” …nan-cial system of continental Europe. The second case would obviously correspond to a situation of absence of agency problems and is consistent with the assump-tion that managers act in the interest of the shareholders. We will simply assume here that the scenario described by our model lies in between the two extreme frameworks.
Note that for the shareholder the yield on shares is given by r¤
s;t= ps;tD¢Nt+
¢ps;t
ps;t ;while for the management the cost of capital is a¤ected by the issuing
value p0; t¢ Nt of the shares. This may be better understood if we accept the
idea that the stock price may diverge in the short run 5 from the value of the
newly installed physical capital and if we admit the possibility of diverging interests between the shareholders and the management.
As we said, the above assumptions are bound to generate not only a recur-sive structure in the problem but also a certain persistence of the past pro…ts in‡uence on the discount rate. The extent of this persistence is limited by the rate of capital depreciation ±: In fact, having de…ned an initial time t = 0 for the beginning of the …rm’s activity, by writing in dynamic terms the ‡ow of funds condition 3, we get: t X i=1 Ii= t X i=1 £
¼i(ki¡1) ¡ ©¤i¡1ki¡1
¤ + t X i=1 ¢Bt or t X i= 1 (ki¡ (1 ¡ ±)ki¡1) = t X i=1 £
¼i(ki¡1) ¡ ©¤i¡1ki¡1
¤ + t X i=1 ¢Bi (8) or again t X i=1 (ki¡ ki¡1) = t X i=1 ¢Ri+ t X i=1 ¢Bi¡ t X i=1 ±ki¡1 (9)
By looking at 8 and 9 it is clear that the ‡ow of pro…ts has a persistent e¤ect on the amount of reserves (i.e. on the share of internally …nanced physically capital). In particular, 9 shows that the degree of persistence is determined by the reciprocal of the rate of capital depreciation 1=±. In this regard we may identify two extreme cases: in the …rst one ± = 1, and in the second one ± = 0:
5or even in the long run according to the ”rational beliefs” theory (see, for instance Kurz
When ± = 1;the capital only lasts for one period and we have kt = £ ¼t(kt¡1) ¡ ©¤t¡1kt¡1 ¤ + Bt; for t > 1:
This means that our problem is still recursive but the impact of the current pro…ts on the …rm’s gearing ratio and discount rate is limited to one period ahead. In this case we would be back to a simpler case of model with intertem-porally nonadditive preferences similar to the one summarized in Obstfeld and Rogo¤ (1996, pp. 722-724) on the basis of Uzawa (1968) .
On the contrary, the assumption ± = 0 correspond to the case of ever-lasting in‡uence of the entire past history of borrowing and pro…ts retention for the determination of the present and future …rm’s discount rate.
In this last case, 9 would become the following:
kt= t X i=1 ¢Ri+ t X i=1 ¢Bi (10) or kt= t X i=1 £
¼i(ki¡1) ¡ ©¤i¡1ki¡1
¤ + t X i=1 ¢Bi= Rt+ Bt
and, having substituted 7 and3 into it, itbecomes the following:
it=
r¤s;t¢ ps;t¢ Nt¡ ¢ps;t¢ Nt
E0+Pti= 1(¼i(ki¡1) ¡ ©¤i¡1ki¡1)
(11)
However let us return now to the more general case where 0 < ± < 1: Equation 5 is the minimum possible weighted average of the borrowed and internally generated …nance. Since the internally generated …nance is pre-determined (by the non-distributed pro…ts at time "t ¡ 1"), by choosing the value ¢Bt the …rms also delimits the maximum amount of feasible new
invest-ments at time "t" and the gearing ratio at time "t", which will be incorporated in the new debt contracts that the …rm issues in order to …nance part of its investments. In other word, as we are going to show later, ¢Bt assumes in this
context the nature of control variable.
To better understand the nature of the problem, it may be useful to …rst develop the minimum value function ©¤
t: Assuming that the second order
con-ditions be satis…ed, the …rst order concon-ditions are the following: d=©¤t=d¹ = rft + Á (¹t) + ¹tÁ0(¹t) ¡ it= 0
this equation (stating that in equilibrium the marginal cost of borrowing equals the marginal cost of the internally generated …nance) can be simpli…ed by assuming that »(¹t) = Á (¹t) + ¹tÁ0(¹t)can be rearranged into a monotonically
increasing and invertible function of ¹t. One can easily verify that this would
be always true if Á (¹t)is convex 6 in ¹
t, which is what we are going to assume.
In this case we would get
¹t = »¡1(it¡ rtf) (12)
This means, in other words, that the gearing ratio is an increasing function of the di¤erence between the cost of own capital it and the interest rate on
risk-free assets rft, since, for a given r f
t, the higher the cost of own capital, the
higher the incentive for the …rm to borrow by increasing the gearing ratio. By looking at the constraints 2 and 4, one immediately sees that they both are dynamic equations putting into relation two ‡ow variables (It and ¢Bt)
with the state variable k at two di¤erent moments in time.(t ¡ 1 and t). In particular, while It relates the state variables kt¡1 and kt for a given rate of
discount ©¤
t; ¢Btdoes the same job and in addition determines (together with kt
) the optimal rate of discount In other words, di¤erently from the conventional neoclassical intertemporal investment models, it is not It but ¢Bt that acts as
a control variable in this context.
Since we know from 4 that ¼t(kt¡1) ¡ It= ©¤t¡1kt¡1¡ ¢Bt , then we may
express 1 in terms of the control variable ¢Bt and the state variable kt¡1 ,
while by putting together the two constraints 4 and 2 we can eliminate It and
express the intertemporal constraints too in terms of ¢Bt: Therefore the …rm’s
problem can be rede…ned as follows:
Vt= (©¤t¡1¢ kt¡1¡ ¢Bt) + 1 X t= 1 ½ (©¤ t ¢ kt¡ ¢Bt+1) (1 + ©¤ t) t ¾ (13) s.t. kt = (1 ¡ ±)kt¡1+ ¼t(kt¡1) ¡ ©¤t¡1kt¡1+ ¢Bt
A brief look at the optimization problem described by 13 and its constrains allows to make a few comments.
On the basis of the assumptions we made about the co-ordination between …nancial and investments decisions and their timing, the stock of capital at time T can be de…ned as follows:
6This would be true also if Á(¹
t) is concave but with a second derivative su¢ciently small
in absolute value, i.e. if its curvature is ”relatively ‡at”. However the assumption of convexity for Á(¹t) is quite general, since it could ”capture” the situation where highly indebted …rms
would have to pay an extremely high risk premium on borrowed capital. Furthermore, if Á (¹t) has an asymptotic behaviour and tends to in…nite when ¹t approaches 1 , one could
reproduce the case of credit rationing by choosing appropriate analytical form and parameters for the function Á (¹t) :
kT = RT(¼T ¡1; kt¡1; ©¤T ¡1; ¼T ¡2; kT ¡2; ©¤T ¡2; :::) + BT(¹¤T).
First of all, if one allows for stochastic shocks in the pro…t function ¼t
(pos-sibly in the variable º , associated to the competitive environment in which the …rm operates, or in the labour market variables) would be transferred to the rate of discount of the future pro…ts from the next period on.
Secondly (and obviously) the share price and its variations also a¤ect the rate of discount (through the dividend policy decided by the management and the consequent cost of own capital)
Finally the very simple model formalisation proposed here may have relevant implications for the empirical phenomenon of countercyclical mark ups. If as suggested by the game-theoretic approach initially proposed by Rotemberg and Saloner (1986), the mark ups are a¤ected by changes (over the business cycle) in the incentives to break the collusion, one of the obvious determinant of the expected discounted ratio of future pro…ts is the ”…rm-speci…c” discount factor, represented here by ©¤
t.
In other words, as we can see from 6, 8, 12 and 5, if we assume that ¼t
follows a cyclical trend and ± is su¢ciently small, even if the share price were strictly correlated to the …rm’s pro…ts, the rate of discount would be slow and sluggish in adjusting to the trend of the …rm’ pro…ts. This implies, in other words, that in phases of high pro…ts and mark ups the …rm might still have a high (and anti-collusive) discount rate determined by the previous negative phases, and, viceversa, at the beginning of the slump, the discount rate might still be low, under the in‡uence of the cheap own capital accumulated in the phases of expansion.
Furthermore, as we can see again from 6, 8, 12 and 5, a change in the …rm’s discount rate (and therefore in the incentives to keep or break the collusion) could be caused by a purely random shock in the share price and in its variation (possibly a su¢ciently persistent speculative bubble). In other words, a random …nancial shock modifying the optimal dividend policy of the …rm’s managers would also modify the cost of own capital, the optimal gearing ratio, and, as a consequence, the discount rate and the incentive of the …rm to collude.
3 A proposed solution approach
Following TU (1991), who formalizes an extension of Pontryagin’s maximum principle to the case of discrete time.(and keeping in mind that we are in a case of intertemporally nonadditive preferences), we can de…ne the discrete Hamiltonian as follows Ht = (©¤t¡1¢ kt¡1¡ ¢Bt) + t+1=±X i=t ( (©¤ i ¢ ki¡ ¢Bi+1) (1 + ©¤ i) i ) + +¸t¡1(¢Bt¡ kt+ (1 ¡ ±)kt¡1+ ¼t(kt¡1) ¡ ©¤t¡1¢ kt¡1
where ©¤ i = ©¤i(¹i(i ¡ r f t))and it = r¤s;t¢ ps;t¢ Nt¡ ¢ps;t¢ Nt
E0+Pti=t¡1=±(¼i(ki¡1) ¡ ©¤i¡1ki¡1)
Assuming now that the regularity conditions for Ht are satis…ed, we have:
@Ht @(¢Bt) = 0 =) ¸t= 1 (14) @Ht @kt¡1 = ¸t which implies @¼t @kt¡1 ¡ ± = = µ @©¤ t @Rt @Rt @(¢Rt) ¶ µ @¼t @kt¡1 ¡ © ¤ t¡1 ¶ ©¤ tkt¡ ¢Bt (1 + ©¤ t)2 ¡ (15) ¡ µ 1 1 + ©¤ t @©¤t @Rt @Rt @(¢Rt) ¶ µ @¼t @kt¡1 ¡ © ¤ t¡1 ¶ ¢ kt+ + t+ 1=± X i=t+1 ½ 1 (1 + ©¤ i)i @©¤ i @Ri @Ri @(¢Rt) µ @¼t @kt¡1 ¡ © ¤ t¡1 ¶ ¢ µ ©¤ iki¡ ¢Bi 1 + ©¤ i ¡ kii ¶¾
Where the three addends on the right-hand side of 15 could be interpreted as follows: The addend @©¤t @Rt @Rt @( ¢Rt) h @¼t @ kt¡1 ¡ © ¤ t¡1 i©¤ tkt¡¢Bt
(1+©¤t)2 can be thought of as the
e¤ect of how the modi…cations
@©¤t @Rt @Rt @ (¢Rt) h @¼t @kt¡1¡ ©¤t¡1 i
in the discount rate generated by a change in the state variable a¤ect the way the future values of (©¤
tkt¡ ¢Bt)are discounted. The addend ¡ 1 1+ ©¤ t @©¤t @ Rt @Rt @ (¢Rt) h @¼t @kt¡1¡ ©¤t¡1 i
¢ kt describes how again the
@©¤t @Rt @Rt @ (¢Rt) h @¼t @kt¡1¡ ©¤t¡1 i
in the discount rate modify the ‡ow of dividends and interest rates that have to be paid on the future capital kt (equal to the
…nancial capital Bt+ Rt ) . The addend t+1=±X i=t+1 ½µ 1 (1 + ©¤ i)i @©¤i @Ri @Ri @(¢Rt) ¶ µ @¼t @kt¡1 ¡ © ¤ t¡1 ¶ µ ©¤iki¡ ¢Bi 1 + ©¤ i ¡ ki ¶¾
jointly represents the two above-mentioned e¤ects for all the future periods in which the change in the rate of discount originated by a change in the state variable still manifest its in‡uence. Equation 15 can be written more compactly in the following way:
@¼t @kt¡1 ¡ ± = µ @¼ t @kt¡1 ¡ ©¤t¡1 ¶X1=± i=t ½µ 1 (1 + ©¤ i)i @©¤i @Ri @Ri @(¢Rt) ¶ µ©¤ iki¡ ¢Bi 1 + ©¤ i ¡ ki ¶¾ (16)
By choosing some speci…c analytical form for all the functions involved in the optimization problem, the result could be seeked recursively from period T backwards.
However a simple look at equation 16 allows to see that the situation of motionless equilibrium, where the marginal pro…tability of capital simply equals the rate of capital depreciation ± , requires the condition ( @¼t
@ kt¡1 ¡ ©
¤
t¡1 = 0),
which also corresponds to the case of exhaustion of pro…ts into dividends and interest rate payments, with no new investments nor increase in borrowing.
4 Conclusions and possible extensions
This paper provides an initial framework to model simultaneity between …rm’s investment and …nancial decisions, assuming diverging incentives for managers and shareholders and …nancial markets imperfections that generate a risk pre-mium on the borrowed …nance.
In such a framework, timing in the process of coordination between …nan-cial and investment decisions is essential for the de…nition of ‡ow variables and might not be properly captured by the conventional continuous time models. For this reason a discrete time recursive structure in the intertemporal problem
of the …rm’s investments has been introduced. This detemines a causal link between the ‡ow of pro…ts, the …rm’s …nancial structure and the rate of discount of the future ‡ows of pro…t. The resulting model implies the endogeneity of the rate of discount in the intertemporal investment problem and a certain lag in its adjustment to the trend followed by the pro…ts of the …rm This particular feature - which might be called ”staggering in the rate of discount” - can repro-duce (according to our conjecture discussed here) the empirical phenomenon of countercyclical mark ups.
Further developments in this line of research can be pursued by assuming that the price share, instead of being exogenous, follows a stochastic path, which might or might not be related to the …rm’s pro…ts according to the assumptions one can make about …nancial markets e¢ciency and about the process of infor-mation spreading that takes place in …nancial markets.
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